The optimization problem defining \(p^*(\theta )\)  has a well-knownThe optimization problem (REF ) appears in [1]}. This is partly a coincidence. Since observation noise is Gaussian, a certain KL divergence is symmetric in its arguments, and our complexity measure aligns with [1]}. More generally, the allocations appearing in [3]}, [4]}, [5]} differ from that in [1]}. solution from the literature on best-arm identification without contexts. See Equation (13) in [1]}. The solution \(p^*\)  is the unique vector satisfying:
\(\frac{ \mu (\theta ,I^*,w) - \mu (\theta ,i,w) }{ \sqrt{ (p^*_{I^*})^{-1}+(p^*_i)^{-1} } } &= \frac{ \mu (\theta ,I^*,w) - \mu (\theta ,j,w) }{ \sqrt{ (p^*_{I^*})^{-1}+(p^*_j)^{-1} } } \qquad \forall i,j \ne I^* \\p_{I^*}^* &= \sqrt{ \sum _{i\ne I^*} (p_i^*)^2 } .\) 
